Question

The tuba typically produces the lowest-frequency sounds in an orchestra. Consider a tuba to be a coiled tube of length \(7.373 \mathrm{~m}\). What is the lowest-frequency sound a tuba can produce? Assume that the speed of sound is \(343.0 \mathrm{~m} / \mathrm{s}\).

Short answer

Answer: The lowest frequency sound a tuba can produce is approximately 23.3 Hz.

Step-by-step solution

Open pipe fundamentals

We have a coiled tube, and we can consider the tube as an open pipe because both ends are open. The formula for the fundamental frequency (lowest frequency) of an open pipe is: \(f_1 = \frac{v}{2L}\) Where: \(f_1\) - the fundamental frequency (Hz) \(v\) - the speed of sound in the air (m/s) \(L\) - the length of the pipe (m) In this case, the length of the tuba, \(L = 7.373 \mathrm{~m}\), and the speed of sound, \(v=343.0 \mathrm{~m} / \mathrm{s}\).

Calculate the fundamental frequency

Now, we substitute the given values into the formula to find the lowest frequency: \(f_1 = \frac{343.0 \mathrm{~m} / \mathrm{s}}{2 \times 7.373 \mathrm{~m}}\) Calculate the denominator first: \(2 \times 7.373 \mathrm{~m} = 14.746 \mathrm{~m}\) Now, divide the numerator by the denominator: \(f_1 = \frac{343.0}{14.746} = 23.251 \mathrm{~Hz}\)

Round the result

The result should be rounded to an appropriate number of significant figures. In this case, we have three significant figures in the given values, so the result should also have three significant figures: Lowest-frequency sound a tuba can produce: \(f_1 \approx 23.3 \mathrm{~Hz}\)

Key concepts

Acoustic Resonance

Acoustic resonance is a phenomenon where an object vibrates in response to external sound waves whose frequency matches the object's natural frequency. Imagine blowing across the top of an empty bottle and hearing a tone. That tone is a result of acoustic resonance, where the air inside the bottle vibrates at a particular frequency.

In musical terms, the tuba, like other wind instruments, makes use of this principle. The air column inside its long, coiled pipe vibrates sympathetically with the player's buzzing lips on the mouthpiece. The wave reflects back and forth within the pipe, and when it constructively interferes with itself at certain frequencies, resonance occurs, producing a rich sound.

Harmonics in Musical Instruments

Harmonics are integral to music, as they contribute to the richness and timbre of the sound produced by musical instruments. Each fundamental frequency, or the lowest frequency at which an instrument can play a note, is accompanied by higher frequencies called overtones or harmonics. These harmonics are integer multiples of the fundamental frequency.

Instruments like the tuba produce a fundamental tone, but the sound we hear contains a mix of many frequencies. The player can modify the pitch by changing the lip tension and by altering the length of the air column, using valves on the instrument, to produce these harmonics.

Speed of Sound

The speed of sound in air, approximately 343 m/s at 20°C, is fundamental to understanding how wind instruments function. The speed at which sound waves travel through a medium like air can be affected by temperature, humidity, and pressure. However, when calculating the fundamental frequency of an instrument such as the tuba, we use a standard value for the speed of sound to maintain uniformity.

It's this speed that determines how quickly the sound waves produced by the instrument reach our ears and the pitch of the sounds that the instrument generates.

Wavelength and Frequency Relationship

The relationship between wavelength and frequency is pivotal when analyzing sounds produced by musical instruments. This relationship is inversely proportional, as represented by the equation:
\[ v = f \lambda \] Where:\( v \) represents the speed of sound, \( f \) is the frequency, and \( \lambda \) is the wavelength of the sound wave. When you increase the frequency of a wave (making the pitch higher), the wavelength becomes shorter, and vice versa.

This relationship is why the tuba, with its long pipe, naturally produces low-frequency sounds. The longer length of the pipe allows for longer wavelengths, which correspond to lower frequencies. Critically, understanding this relationship helps in grasively on the pitch of sounds produced.